Fixing a complete first order theory T, countable for transparency, we had known quite well for which cardinals T has a saturated model. This depends on T of course - mainly of whether it is stable/super-stable. But the older, precursor notion of having a universal notion lead us to more complicated answer, quite partial so far, e.g the strict order property and even SOP_4 lead to having "few cardinals" (a case of GCH almost holds near the cardinal). Note that eg GCH gives a complete
We discuss joint work with Douglas Arnold, Guy David, Marcel Filoche and Svitlana Mayboroda.
Consider for the operator $L = -\Delta + V$ with periodic boundary conditions, and more
generally on the manifold with or without boundary. Anderson localization, a significant feature
of semiconductor physics, says that the eigenfunctions of $L$ are exponentially localized with
high probability for many classes of random potentials $V$. Filoche and Mayboroda introduced the
function $u$ solving $Lu = 1$ and showed numerically that it strongly reflects this localization.